PHYSICAL REVIEW B
VOLUME 58, NUMBER 9
1 SEPTEMBER 1998-I
Method to extract the critical current density and the flux-creep exponent
in high-T c thin films using ac susceptibility measurements
B. J. Jönsson and K. V. Rao
Department of Condensed Matter Physics, Royal Institute of Technology, S-100 44 Stockholm, Sweden
S. H. Yun and U. O. Karlsson
Department of Materials Physics, Royal Institute of Technology, S-100 44 Stockholm, Sweden
~Received 29 December 1997!
High-precision ac susceptibility measurements have been made on high-quality Hg-1212 thin films. A
method to analyze x 8 1 (T,H 0 , f ) and x 9 1 (T,H 0 , f ) and extract the temperature dependence of the critical
current density J c (T), as well as the temperature and field-dependent flux-creep exponent n(T,H 0 ), is presented. With specific measurements at external ac fields H 0 in the range 7 – 100 Oerms we determine the
temperature dependence of the critical current density from a single temperature scan. The obtained temperature dependence, J c (T), is found to be in good agreement with data obtained from measurements using the
traditional ‘‘loss-maximum’’ approach. In addition we present a method to extract the temperature and ac
field-dependent flux-creep exponent n(T,H 0 ) from a set of temperature scans taken at different ac fields and
driving frequencies. The observed power law describing the frequency dependence of x 8 is consistent with a
current-dependent effective activation energy of the form U(J)5U 0 ln(Jc /J). Furthermore, the flux creep is
found to increase with ac field and with temperature except at about 20–30 K below T c , where our data
suggest a slowing down of the flux creep. @S0163-1829~98!02633-2#
I. INTRODUCTION
The vortex dynamics of high-T c superconductors
~HTSC’s! is a complex problem due to the extreme richness
of the magnetic phase diagram.1,2 Interesting functional
properties arise due to the various comparable energy scales
in these materials. The short coherence length j reduces the
pinning energy U pin and the high critical temperature T c
leads to a high thermal energy U th . A direct consequence of
U pin and U th having similar orders of magnitude is the ‘‘giant
flux creep’’ 3 observed in HTSC’s. The penetration length l,
which can be taken as a measure of the range of vortexvortex interactions, is larger than for conventional type-II
low-T c superconductors ~LTSC’s!. As a consequence vortexvortex interactions U int are enhanced and the so-called collective flux-creep phenomenon4 ~CFC! is observed in a wider
field and temperature range compared to LTSC’s. The study
of flux creep in HTSC’s materials is of considerable importance both from a fundamental as well as applications point
of view.
Flux creep manifests itself experimentally in many ways
and there are a number of techniques to probe the vortex
dynamics over a wide range of temperature, magnetic field,
and current density. Three of the most common techniques
are magnetic relaxation, ac susceptibility, and I-V characteristics. The interpretations of results obtained from the different techniques have matured and can now be readily compared. ac susceptibility remains a very popular choice due to
its experimental simplicity: one traditionally determines the
location of the loss maximum T p and studies its dependence
on driving frequency and dc field. It has been pointed out5
that this procedure does not make use of the full available
information, i.e., from an entire temperature or field scan
0163-1829/98/58~9!/5862~6!/$15.00
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only a single point ~the loss maximum location T p ! is extracted. Using square-wave integration,6 instead of the ordinary sine-wave integration, and defining a so-called wideband ac susceptibility7 it has been demonstrated that a single
temperature scan is sufficient to extract the temperature dependence of the critical current density in bulk samples. A
similar procedure using the imaginary part of the ordinary ac
susceptibility has also been suggested8 but to our knowledge
this approach has not been exploited.
We show how recently derived expressions for x 8 (h 0 /J c )
and x 9 (h 0 /J c ) for a thin disk perpendicular to the applied
field9 can be used to extract, not only J c (T) but also the
temperature and ac field-dependent flux-creep exponent
n(T,H 0 ). We test our approach on a high-quality Hg-1212
thin film and compare it with the traditional loss maximum
method. A good agreement is found between the two methods and the efficiency of our method is clearly demonstrated.
In the Hg-based HTSC’s we find that the flux-creep effects
increase ~i.e., n decreases! both with temperature and ac
field. Our data also indicate a slowing down of the flux creep
at about 20–30 K below T c , which might suggest a crossover to another collective flux-creep regime, possibly involving vortex bundles.
II. COMPLEX ac SUSCEPTIBILITY
The interpretations of ac susceptibility data have matured
considerably in the last years as analytical expressions have
been derived for the actual sample and field geometries used
in experiments. The first geometries to be considered were
the infinite cylinder10 and the infinite slab11 in the parallel
applied field where demagnetizing effects can be neglected.
However, for a realistic sample with finite dimensions proper
demagnetizing effects must be taken into account. This is
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© 1998 The American Physical Society
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METHOD TO EXTRACT THE CRITICAL CURRENT . . .
even more so in the case of thin films in a field normal to the
film plan where the demagnetizing effects dominate and the
expressions derived for the parallel geometry fail to describe
the response. The problem of applying the critical-state
model12 ~CS! to the thin circular disk geometry was
solved13,14 in 1993 and subsequently the analytical expressions for the ac susceptibility of thin disks in perpendicular
field were derived.9 The critical state has also been analyzed
for thin long strips15 and squares.16 It is found that due to the
very large demagnetizing effects the particular in-plane
shape almost does not influence the ac response and in particular the ac susceptibility differs by less than 0.2% between
a square and a circular disk over the whole range of applied
fields.17 Unless there is a particular interest in the local character of the flux dynamics it is hence a very good approximation to describe the global response from a rectangular
shaped film with the expressions for a circular disk. This has
recently been verified by experiments on Y-Ba-Cu-O thin
films of different lateral shape.18 A recent numerical study
extends the analytical expressions for the ac susceptibility of
thin films in an ideal CS to thin films exhibiting flux creep.19
Within the framework of the CS model, the ac susceptibility of a thin circular disk with radius R and thickness d in
a perpendicular time-varying field H a (t)5h 0 cos vt has the
limiting large-field behavior9
x 8 '2 x 0 ~ 1.330h 23/220.634h 25/2! , h@1,
~1a!
x 9 ' x 0 ~ h 21 21.059h 22 ! , h@1,
~1b!
where h is the so-called reduced field h52h 0 /J c d and x 0
58R/3p d is the value for complete shielding. In this work
we make explicit use of Eqs. ~1! in the limit where a single
term is sufficient to describe the response,
x 8 '2 x 0 c 1 h 23/2, h@1,
~2a!
x 9 ' x 0 c 2 h 21 , h@1,
~2b!
and show the convenience of such simple relations when flux
creep is studied. It should be noted, however, that Eqs. ~2!
not only apply to a thin circular disk but to any geometry,
provided the applied ac field is large enough for the hysteresis loop to be approximated by a parallelogram.20
III. CRITICAL CURRENT FROM ac SUSCEPTIBILITY
DATA
For many applications the single most important property
of a superconductor is its ability to carry large currents without dissipation. The commonly used21,22 ac susceptibility
technique for measuring the critical current density makes
use of the peak criterion for the imaginary part, which for a
bulk sample in the parallel geometry has the general form
J c ~ T p ! 5 a h 0 /R,
~3!
where a '1 is a geometry-dependent factor ~e.g., 3/4 for
infinite slab, 1 for infinite cylinder!. A similar relation9 holds
for a thin disk in a perpendicular field although due to the
strong demagnetizing effects, not the radius of the sample,
but instead its thickness d enters the expression
J c ~ T p ! 51.03h 0 /d.
~4!
5863
This relation was recently used23 for extracting the temperature dependence of the critical current density in Y-Ba-Cu-O
thin films.
Another, much less used approach makes use of the real
part x 8 of the ac susceptibility.24 In the CS model x 8 is a
single-valued function of the so-called Bean penetration
length L CS5h 0 /J c (T). It is then possible to reconstruct the
temperature dependence of J c (T) by scaling data obtained
for different h 0 .
A more direct approach with simpler functional dependence on L CS is obtained by defining a so-called wide-band
susceptibility7 from the value of the sample magnetization
when the external ac field reaches its maximum ( x a ) and is
zero ( x r ), respectively. In the linear regime the wide-band
susceptibility coincides with the usual complex ac susceptibility but they differ when higher harmonics are present in
the pick-up signal. Due to the simpler functional dependence
of x a (h 0 /J c ) and x r (h 0 /J c ), data from the whole temperature scan can be directly used5 to extract J c (T) instead of
only retaining a single point ~from the loss maximum at T p !
per scan as in the traditional approaches using Eqs. ~3! and
~4!.
We show in this work that it is not necessary to change
the traditional definition of the complex ac susceptibility and
consequently have to change the experimental detection
scheme.6 At high enough ac field amplitudes ~in the following denoted by the root-mean-square value H 0 5h 0 /21/2!, it is
instead possible to use the limiting expressions Eqs. ~2! for
x 8 and x 9 to extract the temperature dependence of the critical current density in superconducting thin films. No scaling
procedure is needed and the full temperature dependence of
J c (T) is obtained from a single temperature scan.
IV. FLUX CREEP FROM ac SUSCEPTIBILITY DATA
It is experimentally known25 that the complex ac susceptibility of high-T c materials can be strongly frequency dependent due to certain time scales in the vortex response.
Experimentally this means that the loss maximum position
T p will not only depend on H 0 through Eq. ~3! but also on
the frequency. The HTSC’s response to an external ac field
can be described by characteristics of one of the following
three different regimes:26 ~i! the CS regime with no detectable frequency dependence but large ac field dependence
(U pin@U th), ~ii! the linear Ohmic regime with eddy current
frequency dependence but no ac field dependence (U pin
!U th), and ~iii! the intermediate flux-creep regime with both
frequency and ac field-dependent ac susceptibility of varying
importance (U pin.'U th). The first two regimes are relatively well understood and can, in principle, be regarded as
limiting cases of the third more complex intermediate regime. A common phenomenological way to describe the
electrodynamics in ~iii! is to assume a nonlinear currentvoltage relation ~nonlinear resistivity!, which in the logarithmic approximation described below takes the form4
r 5E/J5 ~ E c /J c ! u J/J c u n21 .
~5!
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B. J. JÖNSSON, K. V. RAO, S. H. YUN, AND U. O. KARLSSON
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Here n is the so-called flux-creep exponent that determines
the degree of nonlinearity, or equivalently the importance of
flux-creep effects. Small n means large creep effects and the
limiting cases ~i! and ~ii! can be described as n→` and n
51, respectively. For a device exploiting the large critical
current density of superconducting materials it is essential
that n be as large as possible and in the following we will
limit ourselves to the regime with n.1 and therefore do not
further discuss the linear Ohmic regime.
The frequency-dependent loss maximum can be analyzed
if one defines a frequency-dependent critical current
J c ~ T,H 0 , f ! 5J c ~ T,H 0 ! g @ kTln~ f 0 / f ! /U ~ T,H 0 !# ~6!
and applies it to the critical-state formalism.3,24 Here, the
function 0,g(y),1 describes the effective reduction of the
critical current density during one period of the applied ac
field. The functional form of g(y) depends on the detailed
nature of the thermally activated flux motion and in particular in what way the activation energy U depends on the momentary current density. In the so-called logarithmic or Zeldov approximation,27 U depends on current as U(J)
28
5U *
0 ln(Jc /J). Within the collective flux-creep theory
~CFC! such a dependence will turn g(y) into an exponential
g ~ y ! 5exp~ 2y ! .
~7!
Substitution of Eq. ~7! in Eq. ~6! then implies that the critical
current density should depend on frequency following a
power law
J c ~ T,H 0 , f ! 5J c ~ T,H 0 !~ f / f 0 ! 1/n ,
~8!
where n5U(T,H 0 )/kT is the same flux-creep exponent as in
Eq. ~5!. It is thus possible to determine n(T,H 0 ) by measuring J c (T,H 0 , f ) at different ac fields and frequencies, in a
single temperature scan run.
In this work we demonstrate how to extract the flux-creep
exponent n(T,H 0 ) for superconducting thin films from a
small set of temperature scans of x 8 taken at different ac
fields and frequencies. In a typical temperature scan for the
determination of n(T,H 0 ) a single ac field is used and 4–5
different frequencies. The analysis is particularly straightforward due to the simple relation @Eq. ~2a!# between J c and x 8 .
There are in the literature examples of using the frequency
dependence of x 8 to extract n for single-crystal samples.24
However in order to get J c from x 8 those studies use the
rather tedious scaling procedure mentioned above, which
limits its usefulness. We believe that our simpler approach
represents an efficient way to extract the maximum information of the field- and temperature-dependent parameters that
govern the vortex response in high-T c materials.
V. EXPERIMENTAL DETAILS
A conventional two-step method was used for the sample
preparation: deposition of Hg-free precursor films on SrTiO3
substrates followed by annealing at 820 °C for 30 min in a
controlled Hg-vapor atmosphere. Samples were rectangular
in shape with typical area of 234 mm2. Film thickness was
estimated to 4000 Å. The x-ray-diffraction pattern collected
within 5°,2 u ,70° using a Siemens D5000 Diffractometer
FIG. 1. x 8 and x 9 for six different ac fields in the range H 0
53 mOe–30 Oe.
shows mainly lines belonging to the c-axis-oriented Hg-1212
phase with minute traces of c-axis Hg-1223. Rocking curves
on the ~006! line indicate good orientation with full width at
half maximum of around 0.3°. ac susceptibility measurements were carried out in a home-built high sensitivity susceptometer with a three-coil mutual inductance bridge and a
two-position background subtraction scheme. Sine-wave integration on the fundamental frequency was used, i.e., the
susceptometer measures x 81 V and x 91 V ~in units of emu/Oe!,
which in the following will be denoted by x 8 and x 9 for
brevity. Root-mean-square ac fields ranged from H 0
50.1 mOe to 100 Oe and frequencies from 1.81 to 891 Hz.
No dc field was applied and the Earth’s field was not
shielded. The films were always measured perpendicular to
the applied ac field.
VI. RESULTS AND DISCUSSION
A. ac field amplitude dependence—determination of J c „T…
The temperature dependence of the components of the ac
susceptibility x 8 2i x 9 of a rectangular (234 mm2) Hg-1212
thin film is shown in Fig. 1 for increasing applied ac fields
H 0 . At all fields, a single transition is observed as is expected for a high-quality thin film. The transition is broadened as H 0 increases and the loss maximum is suppressed to
lower temperatures. This general behavior is typically expected within a Bean critical-state model. However, we also
observe a weak H 0 dependence of the maximum value of x 9
~in particular for H 0 ,3 Oe! which is not consistent with a
CS description. In this work we do not further analyze this
discrepancy and argue that in order to interpret the experimental results in a CS context it is useful to work at values of
H 0 .3 Oe where a true critical state seems to be established.
In the following we always keep H 0 .7 Oe.
A central issue for the following flux-creep analysis is the
confirmation of the simple relation between J c and x @Eqs.
~2!#. We measured x 8 and x 9 vs T at different ac fields H 0
57 – 100 Oe and studied the ac field dependence of data
taken at same temperatures. For H 0 ,35 Oe we indeed found
a power-law dependence of x 8 and x 9 on H 0 with exponents
21.5 and 21.0, respectively. The validity of these exponents
can be seen by plotting x 8 H 3/2
0 @Figs. 2~a! and 2~b!# and
x 9 H 0 @Figs. 3~a! and 3~b!# vs T, which confirms that over a
large region the different curves overlap. Each set of data
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METHOD TO EXTRACT THE CRITICAL CURRENT . . .
FIG. 2. ~a! Initial part of x 8 for H 0 57, 10, 15, and 20 Oe. ~b!
Plotting x 8 H 3/2
0 vs T makes all data fall onto a single curve in a
certain temperature range DT x 8 (H 0 ) that increases with H 0 . ~c!
1/2.5
vs T yields a straight line.
Plotting ( x 8 H 3/2
0 )
points starts to deviate at an H 0 -dependent temperature as
the second term in Eqs. ~1! becomes comparable to the first.
At ac fields H 0 .35 Oe we observed a slight increase of both
exponents with H 0 . We attribute these stronger field dependences as coming from a field dependence of the critical
current density. Below we will see that this is indeed consistent with a field-dependent flux-creep exponent, n(T,H 0 ).
From the confirmation of Eq. ~2a! we conclude that, in a
certain temperature region, a plot of J c (T)52 3/2H 0
(2 x 8 (T)/c 1 x 0 ) 2/3/d vs T will directly give the temperature
dependence of the critical current density. Assuming a temperature dependence of the form
J c ~ T ! 5J c0 ~ 12T/T c ! b ,
~9!
we expect a straight line if we plot J c (T) 1/b vs T. In practice
this means raising x 8 to some power value 1/n and accepting
the value that yields the best straight line @Fig. 2~c!#. We find
n 52.5 and hence get b 51.7 from Eq. ~2a!. Note that T c
5119.5 K is conveniently found from the intersection of the
5865
FIG. 3. ~a! Initial part of x 9 for H 0 510, 15, 20, 25, 30, and 35
Oe. ~b! All data points fall onto a single curve when plotting
x 9 H 0 vs T. ~c! A straight line is obtained when plotting
( x 9 H 0 ) 1/1.7 vs T.
straight line with the temperature axis. According to Eq. ~2b!
it should be equally valid to extract J c (T) from a single
temperature scan x 9 (T) by finding the exponent 1/n that
transforms x 9 (1/n ) into a linear fit. This is indeed demonstrated by using the same exponent b 51.7 and noting that
over a substantial temperature region all the curves fall onto
the same straight line @Fig. 3~c!#. A closer comparison between the two methods of extracting J c (T) reveals that using
x 8 is the better choice since the temperature region DT
where the data fall on the straight line, i.e., where a single
term in Eqs. ~1! (h 23/2 or h 21 ) dominates the response, is
wider for x 8 than for x 9 . At H 0 520 Oe this range is DT x 8
517 K for x 8 and only DT x 9 58 K for x 9 . For the analysis
in the following section it is important that the data be obtained in the region DT x 8 where we thus can rewrite Eq. ~2a!
as
x 8 ~ T,H 0 , f ! 5constJ c ~ T,H 0 , f ! 3/2,
~10!
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B. J. JÖNSSON, K. V. RAO, S. H. YUN, AND U. O. KARLSSON
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FIG. 4. Temperature dependence of J c obtained using the traditional loss-maximum method. The line is a power-law fit with exponent b 51.7.
and not further consider the prefactor const5
2x0c1(d/2H 0 ) 3/2.
In order to check the validity of this procedure we carried
out a traditional loss maximum determination of the critical
current density vs T and found that a power law with b51.7
represents a good fit to the data ~Fig. 4!. We have thus shown
that a single temperature scan of x 8 at sufficiently high ac
field is enough to determine J c (T). To get the absolute value
of J c (77 K) we use Eq. ~4! ~with h 0 52 1/2H 0 !, approximating our sample with a disk, and find J c (77 K)58.7
3109 A/m2. Extrapolation to lower temperatures using Eq.
~9! yields J c (4.2 K)54.831010 A/m2.
FIG. 6. ~a! log-log plot of x 8 vs f at H 0 5100 Oe. Lines are
power-law fits x 8 (T,H 0 , f )5 P(T,H 0 ) f m ( T,H 0 ) . ~b! same analysis
for H 0 520 Oe with data from the isothermal cuts in Fig. 5.
~11!
which we fit to the data. From these fits we extract the exponent m(T,H 0 ) and then get, via Eq. ~10!, the flux-creep
exponent at all temperatures and different ac fields as
n53/(2m). It is seen in Fig. 7 that m(T,H 0 ) increases
[n(T,H 0 ) decreases# both with temperature and with the applied ac field. The increase of flux creep with temperature is
expected for thermally activated vortex motion. The increase
of flux creep with magnetic field indicates a possible corresponding decrease of the activation energy barrier U(T,H 0 ).
There is also a clear maximum in m(T,H 0 ) @minimum in
n(T,H 0 )# at about 20–30 K below T c and the relaxation
seems to slow down again. It might be that the logarithmic
approximation is no longer valid in this region and that a
new creep regime is entered. Such a slowing down may be
FIG. 5. Initial part of x 8 for H 0 520 Oe and f 51.81, 4.13, 18.1,
181, 891 Hz. The straight lines indicate the isothermal cuts plotted
in Fig. 6~b! below.
FIG. 7. The fitted parameter m(T,H 0 ) as a function of temperature at different ac fields H 0 520, 60, and 100 Oe. On the second
scale is shown the flux-creep exponent n5(3/2)/m.
B. Frequency dependence—determination
of the flux-creep exponent
In Fig. 5 is shown how x 8 increases with frequency,
which indicates the presence of a certain time scale for the
vortex motion. We make isothermal cuts in Fig. 5 and plot
x 8 (T,H 0 , f ) vs f in a log-log diagram ~Fig. 6!. By combining Eqs. ~8! and ~10! we get a power-law expression for
x 8 (T,H 0 , f ),
x 8 ~ T,H 0 , f ! 5 P ~ T,H 0 ! f m ~ T,H 0 ! ,
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METHOD TO EXTRACT THE CRITICAL CURRENT . . .
attributed to flux creep of vortex bundles29 instead of singlevortex creep.
The strong dependence of n and m on H 0 qualitatively
explains the slight deviation from the limiting expressions
@Eq. ~2!# in Sec. VI at the highest ac fields H 0 .35 Oe. At
g
these ac field values x 8 (T,H 0 , f ) goes as H 2
0 , g 51.5– 2
since the reduction with field of
and not exactly as H 23/2
0
J c (T,H 0 , f ) will contribute. From the analysis above we
conclude that this reduction of J c (T,H 0 , f ) with field can be
related to the increase in flux creep for large H 0 .
VII. SUMMARY AND CONCLUSION
We present an ac susceptibility method for the study of
the temperature-dependent critical current density and the
temperature and field-dependent flux-creep exponent. The
main advantage vis-a-vis traditional methods is the amount
of information that can be extracted from a small number of
temperature scans. We show that the results on the critical
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5867
current density agree with results obtained using the traditional ‘‘loss-maximum’’ approach. The critical current density of a high-quality Hg-1212 thin film is well described by
J c (T)5J c0 (12T/T c ) b with b 51.7 in the range 77–120 K.
At 77 K we find J c (77 K)58.73109 A/m2 and an extrapolation to lower temperatures yields J c (4.2 K)54.8
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from the frequency dependence of the real part, x 8 . Flux
creep is found to increase both with temperature and with the
ac field. However, at about 20–30 K below T c a slowing
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crossover to a regime of flux bundle creep.
ACKNOWLEDGMENTS
Fruitful discussions with L. Fabrega and A. M. Grishin
are gratefully acknowledged. This research is supported by
the Swedish Natural Science Researh Council, NFR.
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